**2.1 Modeling of the voltage source**

To model the voltage source (*E0*) of the battery-pack, the relationship between the open circuit voltage (OCV) and the state of charge is calculated by means of current interruption tests. First, the battery-pack is totally charged applying the constant current-constant voltage method (25 A until reaching the maximum voltage). After charging process, the device is discharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time. The OCV value for each test point is recorded when the relaxation time ends. Finally, the batterypack is recharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time as in the case of discharge process. In the same manner, the OCV values are recorded at the end of the relaxation time. The results of these tests are shown in **Figure 2** (discharging test) and **Figure 3** (charging test). **Table 2** presents the OCV values associated with each SOC test point, and this relationship is also sketched in **Figure 4**. As it can be seen, at the end of charge test, the final value of 100% of SOC is not reached because the BMS limits the applied current during the two last pulses. It is important to highlight that this situation does not occur when a single cell is tested, because the cell is charged and discharged from 100% SOC to 0% SOC without protection of BMS. In addition, test results do not show a high deviation of the average values as is reported in literature [25]; therefore, these values are used to evaluate the OCV-SOC relationship, which is presented in Eq. (1).

$$E\_o \text{(SOC,} t) = 26.05 - 0.15 \cdot \text{SOC} + 3.51 \cdot \text{SOC}^2 \tag{1}$$

**147**

**Figure 3.** *Charge test result.*

**Table 2.**

*OCV values at each test point.*

*Hybrid Modeling Procedure of Li-Ion Battery Modules for Reproducing Wide Frequency…*

**Discharge results Charge results**

**OCV SOC (%) OCV SOC (%)** 29.40 100 29.28 90.27 28.77 90 29.06 87.64 28.23 80 28.65 80 27.79 70 28.09 70 27.40 60 27.62 60 26.89 50 27.08 50 26.65 40 26.76 40 26.49 30 26.57 30 26.33 20 26.33 20

*DOI: http://dx.doi.org/10.5772/intechopen.88718*

**2.2 Modeling of the battery-pack's complex impedance**

experimental test bench deeply explained in [26].

To carry out EIS tests, an impedance analyzer is generally used. This device generates a frequency sweep signal and measures the voltage and current in each test point. As a result, the complex impedance is calculated. Because most of commercial impedance analyzer generates AC signals less than 100 mA (suitable for cell testing), in this work, this signal is amplified and controlled by means of the

EIS tests have been performed at different SOC values (20, 40, 60, 80, and 90% SOC) to analyze the effects of these SOC variations. The frequency sweep has been set from 1 mHz to 5 kHz (typically test range), with an AC ripple of 5 A (10% *Imax*). **Figure 5** shows the Nyquist plots of the EIS results, which has been used to analyze the impedance behavior of the tested battery module. In this graph, the real part of the complex impedance (Z') is represented along the x-axis and the imaginary part (Z") along the y-axis. The capacitive behavior corresponds to negative values of Z" and the inductive behavior to the positive ones. In this way, it is easy to identify the parameters of the electrical circuit. According to EIS results, the impedance

**Figure 2.** *Discharge test result.*

*Hybrid Modeling Procedure of Li-Ion Battery Modules for Reproducing Wide Frequency… DOI: http://dx.doi.org/10.5772/intechopen.88718*

#### **Figure 3.** *Charge test result.*

*Research Trends and Challenges in Smart Grids*

**2.1 Modeling of the voltage source**

**Table 1.**

*Battery-pack technical data.*

To model the voltage source (*E0*) of the battery-pack, the relationship between the open circuit voltage (OCV) and the state of charge is calculated by means of current interruption tests. First, the battery-pack is totally charged applying the constant current-constant voltage method (25 A until reaching the maximum voltage). After charging process, the device is discharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time. The OCV value for each test point is recorded when the relaxation time ends. Finally, the batterypack is recharged at current pulses of 10 A for 30 min (0.1 SOC variation) followed by 90 min of relaxation time as in the case of discharge process. In the same manner, the OCV values are recorded at the end of the relaxation time. The results of these tests are shown in **Figure 2** (discharging test) and **Figure 3** (charging test). **Table 2** presents the OCV values associated with each SOC test point, and this relationship is also sketched in **Figure 4**. As it can be seen, at the end of charge test, the final value of 100% of SOC is not reached because the BMS limits the applied current during the two last pulses. It is important to highlight that this situation does not occur when a single cell is tested, because the cell is charged and discharged from 100% SOC to 0% SOC without protection of BMS. In addition, test results do not show a high deviation of the average values as is reported in literature [25]; therefore, these values are used to evaluate the OCV-SOC relationship, which is presented in Eq. (1).

Cells reference MP 176065 Int (Saft batteries)

Pack rated voltage 25.9 V Pack maximum voltage 29.4 V (4.2 V per cell) Pack minimum cutoff voltage 20.3 V (2.9 V per cell) Pack capacity, *Cn* 50 Ah Pack maximum current 50 A Range of temperature (charge) −20 to 60°C Range of temperature (discharge) −30 to 55°C

*Eo* (*SOC*,*t*) = 26.05 − 0.15 ⋅ *SOC* + 3.51 ⋅ *SOC*<sup>2</sup> (1)

**146**

**Figure 2.** *Discharge test result.*


### **Table 2.**

*OCV values at each test point.*
